Finite Group Databasebeta

← #107883

Subgroups — #107883

Order 512.

806 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34399C211yesmaximal, characteristic
2C2×#1559C211yesmaximal, characteristic
3C2×#1485C211yesmaximal, characteristic
4#10858C211yesmaximal, characteristic
5#8035C211yesmaximal, characteristic
6#7914C211yesmaximal, characteristic
7#7899C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C2×#432C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#423C2211yescharacteristic
12C2×C4×M16C2211yescharacteristic
13#2382—22no—
14#2360—22no—
15#2356—21no—
16#2351—21no—
17#2343—22no—
18#2331—21no—
19#2329C414yes—
20#2327—21no—
21#2320—22no—
22#2283C2211yescharacteristic
23C2×#382C2211yescharacteristic
24C2×#343—22no—
25#1559—22no—
26#1559C414yescomplemented
27C2×#333—21no—
28C2×#332—21no—
29#1485—22no—
30#1485C414yescomplemented
31C2×Q8⋊C8—22no—
32#1295—22no—
33#1280—22no—
34#1279—21no—
35#1279—21no—
36C22×M16—21no—
37C22×M16—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41#533—21no—
42#533C2×C412yes—
43#533C2×C412yes—
44#533D412yes—
45#533Q812yes—
46C2×C4⋊Q8D411yescharacteristic
47C2×C4⋊Q8D411yescharacteristic
48C2×#125—21no—
49C2×C22⋊Q8—21no—
50C2×C22⋊Q8C2×C412yes—
51C2×C4×Q8—22no—
52C2×C4×Q8—21no—
53C2×C4×Q8D412yes—
54C2×#117—21no—
55C22×C4⋊C4D412yes—
56C22×C42C2311yescharacteristic
57#432—21no—
58#432—21no—
59#432—21no—
60#432C2×C412yes—
61#431—21no—
62#431—21no—
63#431—21no—
64#431C2×C412yes—
65#423—21no—
66#423—21no—
67#423—21no—
68#423C2×C412yes—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—21no—
73C2×C22⋊C8—21no—
74C2×C22⋊C8—21no—
75C4×M16—21no—
76C4×M16—21no—
77C4×M16—21no—
78C4×M16C2×C412yes—
79C2×#97—21no—
80C2×#97—21no—
81C2×C4×C8—21no—
82C2×C4×C8—21no—
83#391—22no—
84#391—24no—
85#389—24no—
86#389—22no—
87#389—24no—
88#387—24no—
89#387—24no—
90#387—24no—
91#387—24no—
92#387—24no—
93#385—24no—
94#383—22no—
95#383—21no—
96#382—22no—
97#382—21no—
98#382—21no—
99#382—21no—
100#382—21no—
101#382D412yes—
102#381—21no—
103#380C2×C412yes—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4D411yescharacteristic
107C4×C4⋊C4D411yescharacteristic
108C4×C22⋊C4—24no—
109C4×C22⋊C4—21no—
110#376—22no—
111#376—21no—
112C2×#95C2×C412yes—
113C2×#95D412yes—
114#343—44no—
115#333—24no—
116#332—24no—
117Q8⋊C8—28no—
118C22×Q8—21no—
119C22×Q8C22⋊C412yes—
120C23×C4C22⋊C414yes—
121C23×C4C22×C411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D411yescharacteristic
124C2×M16—22no—
125C2×M16—22no—
126C2×M16—22no—
127C2×M16—22no—
128C2×M16—41no—
129C2×M16—41no—
130C2×M16—41no—
131C2×M16—41no—
132C2×M16—42no—
133C2×M16—42no—
134C22×C8—41no—
135C22×C8—41no—
136C22×C8—41no—
137C22×C8—41no—
138C4⋊Q8—21no—
139C4⋊Q8—21no—
140C4⋊Q8—21no—
141C4⋊Q8—21no—
142C4⋊Q8C22⋊C412yes—
143C4⋊Q8C22⋊C412yes—
144C4⋊Q8C4⋊C412yes—
145C4⋊Q8C4⋊C412yes—
146#125—22no—
147#125—22no—
148#125—22no—
149#125—22no—
150C22⋊Q8—22no—
151C22⋊Q8—22no—
152C22⋊Q8—22no—
153C22⋊Q8—22no—
154C22⋊Q8—22no—
155C22⋊Q8—22no—
156C22⋊Q8C4214yes—
157C22⋊Q8C4⋊C414yes—
158C4×Q8—22no—
159C4×Q8—22no—
160C4×Q8—22no—
161C4×Q8—22no—
162C4×Q8—22no—
163C4×Q8—22no—
164C4×Q8—82no—
165C4×Q8—82no—
166C4×Q8C22⋊C414yes—
167C4×Q8C4⋊C414yes—
168#117—22no—
169#117—22no—
170#117—22no—
171#117—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4C22⋊C412yes—
184C2×C4⋊C4C22⋊C412yes—
185C2×C4⋊C4C22⋊C412yes—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C42—22no—
189C2×C42—21no—
190C2×C42—21no—
191C2×C42—24no—
192C2×C42—24no—
193C2×C42C22×C411yescharacteristic
194C2×C42C22×C411yescharacteristic
195C2×C42C2×D411yescharacteristic
196C2×C42C2×D411yescharacteristic
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D411yescharacteristic
199C2×C42C2×D411yescharacteristic
200C2×C42C2×Q811yescharacteristic
201C2×C42#4111yescharacteristic
202C2×C42#4111yescharacteristic
203C2×C42#4111yescharacteristic
204C2×C42#4111yescharacteristic
205C4⋊C8—22no—
206C4⋊C8—22no—
207C4⋊C8—22no—
208C4⋊C8—22no—
209C4⋊C8—41no—
210C4⋊C8—41no—
211C4⋊C8—41no—
212C4⋊C8—41no—
213C22⋊C8—22no—
214C22⋊C8—22no—
215C22⋊C8—41no—
216C22⋊C8—41no—
217#97—22no—
218#97—22no—
219#97—41no—
220#97—41no—
221C4×C8—22no—
222C4×C8—22no—
223C4×C8—41no—
224C4×C8—41no—
225#95—24no—
226#95—24no—
227#95—24no—
228#95—24no—
229#95—22no—
230#95—22no—
231#95—22no—
232#95—22no—
233#95—22no—
234#95—21no—
235#95—22no—
236#95—22no—
237#95—21no—
238C24C2×C22⋊C411yescharacteristic
239C2×Q8—22no—
240C2×Q8—22no—
241C2×Q8—22no—
242C2×Q8—44no—
243C2×Q8—41no—
244C2×Q8—41no—
245C2×Q8—41no—
246C2×Q8—41no—
247C2×Q8#9514yes—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—24no—
256C22×C4—22no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—44no—
260C22×C4—44no—
261C22×C4—44no—
262C22×C4C2×C4211yescharacteristic
263C22×C4C2×C22⋊C411yescharacteristic
264C22×C4C2×C22⋊C411yescharacteristic
265C22×C4C2×C4⋊C411yescharacteristic
266C22×C4#11711yescharacteristic
267C22×C4#11711yescharacteristic
268C22×C4C4×D411yescharacteristic
269C22×C4C4×D411yescharacteristic
270C22×C4C4×D411yescharacteristic
271C22×C4C4×D411yescharacteristic
272C22×C4#12011yescharacteristic
273C22×C4#12011yescharacteristic
274C22×C4#12011yescharacteristic
275C22×C4#12011yescharacteristic
276C22×C4#12111yescharacteristic
277C22×C4#12111yescharacteristic
278C22×C4C22⋊Q811yescharacteristic
279C22×C4C22⋊Q811yescharacteristic
280M16—24no—
281M16—42no—
282M16—42no—
283M16—42no—
284M16—44no—
285M16—82no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—82no—
293C2×C8—82no—
294C2×C8—81no—
295C2×C8—81no—
296C4⋊C4—24no—
297C4⋊C4—24no—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4—24no—
303C4⋊C4—22no—
304C4⋊C4—22no—
305C4⋊C4—22no—
306C4⋊C4—24no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—82no—
312C4⋊C4—84no—
313C4⋊C4—84no—
314C4⋊C4#9514yes—
315C22⋊C4—24no—
316C22⋊C4—24no—
317C22⋊C4—44no—
318C42—24no—
319C42—24no—
320C42—41no—
321C42—41no—
322C42—41no—
323C42—41no—
324C42—84no—
325C42—84no—
326C42C2×C22⋊C411yescharacteristic
327C42C2×C22⋊C411yescharacteristic
328C42C2×C4⋊C411yescharacteristic
329C42C2×C4⋊C411yescharacteristic
330C42C4×D411yescharacteristic
331C42C4×D411yescharacteristic
332C42C4×D411yescharacteristic
333C42C4×D411yescharacteristic
334C42#12111yescharacteristic
335C42#12111yescharacteristic
336C42C22⋊Q811yescharacteristic
337C42C22⋊Q811yescharacteristic
338C42#12311yescharacteristic
339C42#12311yescharacteristic
340C42#12311yescharacteristic
341C42#12311yescharacteristic
342C23—21no—
343C23—21no—
344C23—22no—
345C23—22no—
346C23C4×C22⋊C411yescharacteristic
347C23#37911yescharacteristic
348C23#38011yescharacteristic
349Q8—44no—
350Q8—42no—
351Q8—42no—
352Q8—42no—
353Q8—42no—
354Q8—82no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—44no—
382C2×C4—42no—
383C2×C4—44no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—44no—
388C2×C4—42no—
389C2×C4—44no—
390C2×C4—42no—
391C2×C4—164no—
392C2×C4C2×#9511yescharacteristic
393C2×C4C4×C22⋊C411yescharacteristic
394C2×C4#38011yescharacteristic
395C2×C4#38111yescharacteristic
396C2×C4#38511yescharacteristic
397C2×C4#38511yescharacteristic
398C2×C4#38511yescharacteristic
399C2×C4#38511yescharacteristic
400C2×C4#38611yescharacteristic
401C2×C4#38611yescharacteristic
402C2×C4#38611yescharacteristic
403C2×C4#38611yescharacteristic
404C2×C4#45311yescharacteristic
405C2×C4#45411yescharacteristic
406C2×C4#45511yescharacteristic
407C2×C4#45611yescharacteristic
408C8—44no—
409C8—82no—
410C8—82no—
411C8—82no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—42no—
419C22—42no—
420C22C22⋊#9511yescharacteristic
421C22#179112yes—
422C22#179211yescharacteristic
423C22#179311yescharacteristic
424C22#179612yes—
425C4—21no—
426C4—21no—
427C4—21no—
428C4—21no—
429C4—21no—
430C4—21no—
431C4—21no—
432C4—21no—
433C4—21no—
434C4—21no—
435C4—44no—
436C4—44no—
437C4—82no—
438C4—84no—
439C4—82no—
440C4#185911yescharacteristic
441C4#186011yescharacteristic
442C4#188211yescharacteristic
443C4#188311yescharacteristic
444C2—41no—
445C2—41no—
446C2#848412yes—
447C2#848512yes—
448C2#1074711yescharacteristic
449C2#1077711yescharacteristic
450C2#1078111yescharacteristic
subgroupG/H
center, socleC23#379
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
26. #1559C4semidirect
30. #1485C4semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#343992
C2×#15592
C2×#14852
#108582
#80352
#79142
#78992